Let be a given infinite series with sequence of partial sums . Let denote the sequence of means of  . The transform of is defined by

where

Necessary and sufficient conditions for the method to be regular are

(i) for each ,

(ii), where is a positive constant independent of .

The series is said to be summable , , if

where

where as .

The series is said to be summable , if

where

and it is said to be summable , , if

where is as defined by (1.1).

For , summability reduces to summability.

The series is said to be bounded or if

By , we denote the set of sequences satisfying

It is known (Das [1]) that for , (1.5) holds if and only if

For , the series is said to be -summable, , (Sulaiman [2]), if

where as .

It is quite reasonable to give the following definition.

For , the series is said to be -summable, , if

where as .

We also assume that , are positive sequences of numbers such that

A positive sequence is said to be a quasi--power increasing sequence, , if there exists a constant such that

holds for (see [3]).

Das [1], in 1966, proved the following result.

Theorem 1.1.

Let , . Then if is -summable, it is -summable.

Recently Singh and Sharma [4] proved the following theorem.

Theorem 1.2.

Let , and let be a monotonic nondecreasing sequence for . The necessary and sufficient condition that is -summable whenever

is that

2. Lemmas

Lemma 2.1.

Let be nonincreasing, . Then for , ,

Proof.

Since is nonincreasing, then .

Therefore

Lemma 2.2.

For ,

Proof.

Since ,  then is nonincreasing and hence

Lemma 2.3 (see [3]).

If is a quasi-f-increasing sequence, where , , , then under the conditions

one has

3. Result

Our aim is to present the following new general result.

Theorem 3.1.

Let , and let be a quasi-f-increasing sequence, where , , and (2.6), and

are all satisfied, then the series is summable , .

Proof.

We have

In order to prove the result, it is sufficient, by Minkowski's inequality, to show that

Applying HÖlder's inequality, we have

This completes the proof of the theorem.